introduction and mathematical concepts
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Introduction and Mathematical Concepts. Chapter 1. 1.1 The Nature of Physics. Physics has developed out of the efforts of men and women to explain our physical environment. Physics encompasses a remarkable variety of phenomena: planetary orbits radio and TV waves magnetism lasers - PowerPoint PPT PresentationTRANSCRIPT
Introduction and Mathematical Concepts
Chapter 1
1.1 The Nature of Physics
Physics has developed out of the effortsof men and women to explain our physicalenvironment.
Physics encompasses a remarkable variety of phenomena:
planetary orbitsradio and TV wavesmagnetismlasers
many more!
1.1 The Nature of Physics
Physics predicts how nature will behavein one situation based on the results of experimental data obtained in anothersituation.
Newton’s Laws → Rocketry
Maxwell’s Equations → Telecommunications
1.2 Units
Physics experiments involve the measurementof a variety of quantities.
These measurements should be accurate andreproducible.
The first step in ensuring accuracy andreproducibility is defining the units in whichthe measurements are made.
1.2 Units
SI units
meter (m): unit of length
kilogram (kg): unit of mass
second (s): unit of time
1.2 Units
1.2 Units
1.2 Units
1.2 Units
The units for length, mass, and time (aswell as a few others), are regarded asbase SI units.
These units are used in combination to define additional units for other important
physical quantities such as force and energy.
1.3 The Role of Units in Problem Solving
THE CONVERSION OF UNITS
1 ft = 0.3048 m
1 mi = 1.609 km
1 hp = 746 W
1 liter = 10-3 m3
1.3 The Role of Units in Problem Solving
Example 1 The World’s Highest Waterfall
The highest waterfall in the world is Angel Falls in Venezuela,with a total drop of 979.0 m. Express this drop in feet.
Since 3.281 feet = 1 meter, it follows that
(3.281 feet)/(1 meter) = 1
feet 3212meter 1
feet 281.3meters 0.979 Length
1.3 The Role of Units in Problem Solving
1.3 The Role of Units in Problem Solving
Reasoning Strategy: Converting Between Units
1. In all calculations, write down the units explicitly.
2. Treat all units as algebraic quantities. When identical units are divided, they are eliminated algebraically.
3. Use the conversion factors located on the pagefacing the inside cover. Be guided by the fact that multiplying or dividing an equation by a factor of 1does not alter the equation.
1.3 The Role of Units in Problem Solving
Example 2 Interstate Speed Limit
Express the speed limit of 65 miles/hour in terms of meters/second.
Use 5280 feet = 1 mile and 3600 seconds = 1 hour and 3.281 feet = 1 meter.
second
feet95
s 3600
hour 1
mile
feet 5280
hour
miles 6511
hour
miles 65 Speed
second
meters29
feet 3.281
meter 1
second
feet951
second
feet95 Speed
1.3 The Role of Units in Problem Solving
DIMENSIONAL ANALYSIS
[L] = length [M] = mass [T] = time
221 vtx
Is the following equation dimensionally correct?
TLTT
LL 2
1.3 The Role of Units in Problem Solving
Is the following equation dimensionally correct?
vtx
LTT
LL
1.4 Trigonometry
1.4 Trigonometry
h
hosin
h
hacos
a
o
h
htan
1.4 Trigonometry
m2.6750tan oh
m0.80m2.6750tan oh
a
o
h
htan
1.4 Trigonometry
h
ho1sin
h
ha1cos
a
o
h
h1tan
1.4 Trigonometry
a
o
h
h1tan 13.9m0.14
m25.2tan 1
1.4 Trigonometry
222ao hhh Pythagorean theorem:
1.5 Scalars and Vectors
A scalar quantity is one that can be describedby a single number:
temperature, speed, mass
A vector quantity deals inherently with both magnitude and direction:
velocity, force, displacement
1.5 Scalars and Vectors
By convention, the length of a vectorarrow is proportional to the magnitudeof the vector.
8 lb4 lb
Arrows are used to represent vectors. Thedirection of the arrow gives the direction ofthe vector.
1.5 Scalars and Vectors
1.6 Vector Addition and Subtraction
Often it is necessary to add one vector to another.
1.6 Vector Addition and Subtraction
5 m 3 m
8 m
1.6 Vector Addition and Subtraction
1.6 Vector Addition and Subtraction
2.00 m
6.00 m
1.6 Vector Addition and Subtraction
2.00 m
6.00 m
222 m 00.6m 00.2 R
R
m32.6m 00.6m 00.2 22 R
1.6 Vector Addition and Subtraction
2.00 m
6.00 m
6.32 m
00.600.2tan
4.1800.600.2tan 1
1.6 Vector Addition and Subtraction
When a vector is multiplied by -1, the magnitude of the vector remains the same, but the direction of the vector is reversed.
1.6 Vector Addition and Subtraction
A
B
BA
A
B
BA
1.7 The Components of a Vector
. ofcomponent vector theand
component vector thecalled are and
r
yx
y
x
1.7 The Components of a Vector
.AAA
AA
A
yx
that soy vectoriall together add and
axes, and the toparallel are that and vectors
larperpendicu twoare of components vector The
yxyx
1.7 The Components of a Vector
It is often easier to work with the scalar components rather than the vector components.
. of
componentsscalar theare and
A
yx AA
1. magnitude with rsunit vecto are ˆ and ˆ yx
yxA ˆˆ yx AA
1.7 The Components of a Vector
Example
A displacement vector has a magnitude of 175 m and points atan angle of 50.0 degrees relative to the x axis. Find the x and ycomponents of this vector.
rysin
m 1340.50sinm 175sin ry
rxcos
m 1120.50cosm 175cos rx
yxr ˆm 134ˆm 112
1.8 Addition of Vectors by Means of Components
BAC
yxA ˆˆ yx AA
yxB ˆˆ yx BB
1.8 Addition of Vectors by Means of Components
yx
yxyxC
ˆˆ
ˆˆˆˆ
yyxx
yxyx
BABA
BBAA
xxx BAC yyy BAC